We know that a classification of manifolds include:
(TOP) topological manifolds
(PDIFF), for piecewise differentiable
(PL) piecewise-linear manifolds
(DIFF) the smooth manifolds
I know that
1) Diff (Differentiable) manifold = smooth manifold
2) Diff manifold is always a triangulable manifold?
question 1: Is Diff manifold always uniquely triangulated? What do we mean by uniquely triangulated?
3) PL manifold = Diff manifold up to 6 dimensions.
4) PL manifold in 7 dimensions may be smoothed=differentiable, but not uniquely.
question 2: Is PL manifold in 7 dimensions still triangulable? (although some triangulation may not be unique?)
question 3: Thus PL manifold may not be triangulable, in which dimensions (such as 7 or 8 dimensions, and/or above)?